---
title: "Modularity and effective Mordell I"
authors:
  - "Levent Alpoge"
year: 2021
venue: "arXiv preprint"
arxiv: "2109.07917"
doi: null
source: "https://arxiv.org/abs/2109.07917"
topics:
  - "arithmetic-geometry"
  - "number-theory"
seed_rank: 1206
seed_batch: "alpoge-math-2026-08-24"
reviewed: "2026-08-24"
pool: "maths-foundations"
relevance_score: 9
lineage: arithmetic-geometry
cites:
  - title: "Un peu d'effectivite pour les varietes modulaires de Hilbert-Blumenthal"
    url: "https://arxiv.org/abs/2111.12466"
    year: 2021
    arxiv: "2111.12466"
    doi: null
see:
  - "1203-un-peu-d-effectivite-pour-les-varietes-modulaires-de-hilbert-b"
---

# Modularity and effective Mordell I

## One-sentence takeaway

Effective Faltings for curves that map to Hilbert modular stacks over odd-degree totally real fields, via effective Shafarevich for GL_2-type abelian varieties over those fields — including height bounds on C_a: x^6+4y^3=a^2.

## Why it's lovely

Unconditional effectivity in a modular family. Over Q-bar every hyperbolic hyperelliptic curve has an etale cover dominating C_1.

## Problem

Faltings is ineffective. Need a modularity handle.

## Design

Effective Shafarevich for GL_2-type over odd-degree totally real K, then pull back to the curves.

## Evidence

arXiv:2109.07917, v2 24 Nov 2021.

## Limitations

Odd-degree totally real. Part I; no Part II on arXiv as of ingest.

## Implications for Broadside

The unconditional special case 1200 dreamed of. See 1203.

## Bottom line

Mint 2109.07917.

## Links

- arXiv: [2109.07917](https://arxiv.org/abs/2109.07917)